TEXTBOOK NOTES – MODULE 5
THE DERIVATIVE OF A PRODUCT
•
Find the Error: Find the error in the following computation of the derivative of 𝑓 (𝑥 ) = 𝑥 2 + 3𝑥. What is
the correct derivative of 𝑓(𝑥 ) in this case?
𝑑 2
[𝑥 + 3𝑥 ]
𝑑𝑥
𝑑
[𝑥 (𝑥 + 3)]
=
𝑑𝑥
𝑑
𝑑
[𝑥 ] ⋅
[𝑥 + 3]
=
𝑑𝑥
𝑑𝑥
= (1) ⋅ (1)
𝑓 ′ (𝑥 ) =
= 1
•
The Product Rule: If 𝑓 and 𝑔 are differentiable functions, then
𝑑
(𝑓(𝑥 ) ⋅ 𝑔(𝑥 )) =
𝑑𝑥
THE DERIVATIVE OF A QUOTIENT
•
The Quotient Rule: If 𝑓 and 𝑔 are differentiable functions, then
𝑑 𝑓 (𝑥 )
(
)=
𝑑𝑥 𝑔(𝑥 )
•
Business Connections: Complete the table and statements below with information connecting
computations with quotients and derivatives to business quantities.
Quantity Description
Average Cost Function
Average Revenue Function
Average Profit Function
Input Description Output Description Function Formula
▪
The Marginal Average Cost/Revenue/Profit Function
•
The marginal average cost function is…
•
The marginal average revenue function is…
•
The marginal average profit function is…
THE CHAIN RULE
•
•
Identify the Formula: Give two forms of the formula for the derivative of a composite function
𝑦 = 𝑓 [𝑔(𝑥 )], assuming all associated derivatives exist. (Hint: Check page 336 of the text!)
▪
Original Form:
▪
Alternate Form:
Identify the Formula: Complete the table below with applications of the Chain Rule to several selected
types of functions.
Rule Name
Original Function Derivative Function
Generalized Power Rule
𝑦 = (𝑔(𝑥 ))
𝑛
Generalized Exponential (base 𝑒) Rule 𝑦 = 𝑒 𝑔(𝑥)
Generalized Exponential (base 𝑏) Rule 𝑦 = 𝑏 𝑔(𝑥)
Generalized Natural Logarithm Rule
𝑦 = ln(𝑔(𝑥 ))
Generalized Logarithm (base 𝑏) Rule
𝑦 = log 𝑏 (𝑔(𝑥 ))